Analysis and algebra on differentiable manifolds : a workbook for students and teachers
Title:
Analysis and algebra on differentiable manifolds : a workbook for students and teachers
Author:
Gadea, P. M. (Pedro Martínez)
ISBN:
9789400759510
Personal Author:
Edition:
2nd ed.
Publication Information:
London ; New York : Springer, ©2013.
Physical Description:
xxv, 617 pages : illustrations (some color) ; 24 cm.
Series:
Problem books in mathematics
Problem books in mathematics.
Contents:
Differentiable Manifolds. Some Definitions and Theorems on Differentiable Manifolds ; Canifolds ; Differentiable Structures Defined on Sets ; Differentiable Functions and Mappings ; CriticalPoints andValues ; Immersions, Submanifolds, Embeddings and Diffeomorphisms ; Constructing Manifolds by Inverse Image. Implicit Map Theorem ; Submersions. Quotient Manifolds ; The Tangent Bundle ; VectorFields. -- Tensor Fields and Differential Forms. Some Definitions and Theorems on Tensor Fields and Differential Forms ; Vector Bundles ; Tensor and Exterior Algebras. Tensor Fields ; Differential Forms. Exterior Product ; Lie Derivative. Interior Product ; Distributions and Integral Manifolds. Frobenius Theorem. Differential Ideals. -- Integration on Manifolds. Some Definitions and Theorems on Integration on Manifolds ; OrientableManifolds. Orientation-PreservingMaps ; Integration on Chains. Stokes' Theorem I ; Integration on Oriented Manifolds. Stokes; Theorem II ; De Rham Cohomology--^ Lie Groups. Some Definitions and Theorems on Lie Groups ; Lie Groups and Lie Algebras ; Homomorphisms of Lie Groups and Lie Algebras ; Lie Subgroups and Lie Subalgebras ; The Exponential Map ; TheAdjointRepresentation ; Lie Groups of Transformations ; Homogeneous Spaces. -- Fibre Bundles. Some Definitions and Theorems on Fibre Bundles ; Principal Bundles ; Connections in Bundles ; CharacteristicClasses ; Linear Connections ; TorsionandCurvature ; Geodesics ; AlmostComplexManifolds ; AlmostSymplecticManifolds--^ Riemannian Geometry. Some Definitions and Theorems on Riemannian Geometry ; Riemannian Manifolds ; Riemannian Connections ; Geodesics ; The Exponential Map ; Gauss'Lemma forCovariantSymmetricTensors ; Curvature andRicciTensors ; CharacteristicClasses ; Isometries ; Left-Invariant Metrics on Lie Groups ; Homogeneous Riemannian Manifolds and Riemannian Symmetric Spaces ; Spaces of Constant Curvature ; Gradient, Divergence, Codifferential, Curl, Laplacian, and Hodge Star Operator on Riemannian Manifolds ; Affine, Killing, Conformal, Projective, Jacobi, and Harmonic VectorFields ; Submanifolds. Second Fundamental Form ; EnergyofHopfVectorFields ; Surfaces in R3 ; Pseudo-Riemannian Manifolds. -- Some Formulas and Tables.
Abstract:
"This is the second edition of this best selling problem book for students, now containing over 400 completely solved exercises on differentiable manifolds, Lie theory, fibre bundles and Riemannian manifolds. The exercises go from elementary computations to rather sophisticated tools. Many of the definitions and theorems used throughout are explained in the first section of each chapter where they appear. A 56-page collection of formulae is included which can be useful as an aide-mémoire, even for teachers and researchers on those topics. This book will be useful to advanced undergraduate and graduate students of mathematics, theoretical physics and some branches of engineering with a rudimentary knowledge of linear and multilinear algebra."--Publisher's website.
Subject Term: